# msr

##### Signed or Vector-Valued Measure

Defines an object representing a signed measure or vector-valued measure on a spatial domain.

##### Usage

`msr(qscheme, discrete, density, check=TRUE)`

##### Arguments

- qscheme
A quadrature scheme (object of class

`"quad"`

usually extracted from a fitted point process model).- discrete
Vector or matrix containing the values (masses) of the discrete component of the measure, for each of the data points in

`qscheme`

.- density
Vector or matrix containing values of the density of the diffuse component of the measure, for each of the quadrature points in

`qscheme`

.- check
Logical. Whether to check validity of the arguments.

##### Details

This function creates an object that represents a
signed or vector valued *measure* on the two-dimensional plane.
It is not normally called directly by the user.

A signed measure is a classical mathematical object (Diestel and Uhl, 1977) which can be visualised as a collection of electric charges, positive and/or negative, spread over the plane. Electric charges may be concentrated at specific points (atoms), or spread diffusely over a region.

An object of class `"msr"`

represents a signed (i.e. real-valued)
or vector-valued measure in the spatstat package.

Spatial residuals for point process models
(Baddeley et al, 2005, 2008) take the form of a real-valued
or vector-valued measure. The function
`residuals.ppm`

returns an object of
class `"msr"`

representing the residual measure.
Various other diagnostic tools such as `dfbetas.ppm`

and
`dffit.ppm`

also return an object of class `"msr"`

.

The function `msr`

would not normally be called directly by the
user. It is the low-level creator function that
makes an object of class `"msr"`

from raw data.

The first argument `qscheme`

is a quadrature scheme (object of
class `"quad"`

). It is typically created by `quadscheme`

or
extracted from a fitted point process model using
`quad.ppm`

. A quadrature scheme contains both data points
and dummy points. The data points of `qscheme`

are used as the locations
of the atoms of the measure. All quadrature points
(i.e. both data points and dummy points)
of `qscheme`

are used as sampling points for the density
of the continuous component of the measure.

The argument `discrete`

gives the values of the
atomic component of the measure for each *data point* in `qscheme`

.
It should be either a numeric vector with one entry for each
data point, or a numeric matrix with one row
for each data point.

The argument `density`

gives the values of the *density*
of the diffuse component of the measure, at each
*quadrature point* in `qscheme`

.
It should be either a numeric vector with one entry for each
quadrature point, or a numeric matrix with one row
for each quadrature point.

If both `discrete`

and `density`

are vectors
(or one-column matrices) then the result is a signed (real-valued) measure.
Otherwise, the result is a vector-valued measure, with the dimension
of the vector space being determined by the number of columns
in the matrices `discrete`

and/or `density`

.
(If one of these is a \(k\)-column matrix and the other
is a 1-column matrix, then the latter is replicated to \(k\) columns).

The class `"msr"`

has methods for `print`

,
`plot`

and `[`

.
There is also a function `Smooth.msr`

for smoothing a measure.

##### Value

An object of class `"msr"`

.

##### Guide to using measures

Objects of class `"msr"`

, representing measures, are returned by the
functions `residuals.ppm`

, `dfbetas.ppm`

,
`dffit.ppm`

and possibly by other functions.

There are methods for printing and plotting a measure, along with
many other operations, which can be listed by
typing `methods(class="msr")`

.

The `print`

and `summary`

methods report basic information
about a measure, such as the total value of the measure, and the
spatial domain on which it is defined.

The `plot`

method displays the measure. It is documented separately in
`plot.msr`

.

A measure can be smoothed using `Smooth.msr`

, yielding a
pixel image which is sometimes easier to interpret than the plot
of the measure itself.

The subset operator `[`

can be used to restrict the measure to
a subregion of space, or to extract one of the scalar components of a
vector-valued measure. It is documented separately in
`[.msr`

.

The total value of a measure, or the value on a subregion,
can be obtained using `integral.msr`

.
The value of a measure `m`

on a subregion
`B`

can be obtained by `integral(m, domain=B)`

or
`integral(m[B])`

. The values of a measure `m`

on each tile
of a tessellation `A`

can be obtained by
`integral(m, domain=A)`

.

Some mathematical operations on measures are supported, such as multiplying a measure by a single number, or adding two measures.

Measures can be separated into components in different ways using
`as.layered.msr`

, `unstack.msr`

and `split.msr`

.

Internal components of the data structure of an `"msr"`

object
can be extracted using `with.msr`

.

##### References

Baddeley, A., Turner, R., Moller, J.
and Hazelton, M. (2005)
Residual analysis for spatial point processes.
*Journal of the Royal Statistical Society, Series B*
**67**, 617--666.

Baddeley, A., Moller, J.
and Pakes, A.G. (2008)
Properties of residuals for spatial point processes.
*Annals of the Institute of Statistical Mathematics*
**60**, 627--649.

Diestel, J. and Uhl, J.J. Jr (1977)
*Vector measures*.
Providence, RI, USA: American Mathematical Society.

Halmos, P.R. (1950) *Measure Theory*. Van Nostrand.

##### See Also

`plot.msr`

,
`Smooth.msr`

,
`[.msr`

,
`with.msr`

,
`split.msr`

,
`Ops.msr`

,
`measureVariation`

,
`measureContinuous`

.

##### Examples

```
# NOT RUN {
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
rp <- residuals(fit, type="pearson")
rp
rs <- residuals(fit, type="score")
rs
colnames(rs)
# An equivalent way to construct the Pearson residual measure by hand
Q <- quad.ppm(fit)
lambda <- fitted(fit)
slam <- sqrt(lambda)
Z <- is.data(Q)
m <- msr(Q, discrete=1/slam[Z], density = -slam)
m
# }
```

*Documentation reproduced from package spatstat, version 1.59-0, License: GPL (>= 2)*